Research

1.   Numerical Methods for Moving Interface and Free Boundary Problems.

Moving interface and free boundary occur in many multiphase and multi-physics problems that arise in science and engineering. Two main numerical challenges are:

1) representing and tracking a moving interface that is highly dynamical, deformable and with possible topological changes,

2) solving global dynamics with interface condition, i.e., solving partial different equations with jump and/or flux conditions at the interface.

I am interested in both issues and applications.

1.   Inverse Problem and Imaging.

In general inverse problems are ill-posedness and nonlinear (even the forward problem is linear).  Ill-posedness means sensitive to perturbation, such as noise. Nonlinearity means iterative construction are usually needed. These two natures of inverse problems posing great challenge to numerical algorithms.

I am interested in imaging problems such as CT and MRI, optical and molecular imaging, wave propagation and seismic, travel time tomography and inverse scattering.

1.   Image Processing and Computer Vision.

Point cloud is a basic and ubiquitous form of data for 3D modeling. I am interested in processing, visualization, and surface reconstruction from 3D point cloud. I am also interested in data analysis and manifold learning for high dimensional data. Image processing and medical imaging are also of great interest to me.

1.   Hamilton-Jacobi Equation and the Fast Sweeping Method (FSM).

Hamilton-Jacobi equation is a type of nonlinear hyperbolic partial differential equation that appears in control, game theory, classical mechanics, geometric optics, and many other applications. Classical solution does not exit in general and appropriate weak solution, the viscosity solution, has to be defined. Fast Sweeping Method (FSM) is an efficient iterative method that can solve hyperbolic type of partial differential equations (PDE) with optimal complexity. The key point is to combine upwind scheme with causality enforced Gauss-Seidel iteration and systematic ordering that covers all possible directions of characteristics. We have developed efficient algorithms for general convex Hamilton-Jacobi equations on structured or unstructured mesh in 2D or 3D.

Software Development

Publications by topics

Moving Interface and Free Boundary Problems

1. 1. A Multi-Scale Method for Dynamics Simulation in Continuum Solvents I: Finite-Difference Algorithm for Navier-Stokes Equation, L. Xiao, Q. Cai, Z. Li, H. Zhao and R. Luo, Chemical Physics Letters,Chemical Physics Letters, 616, 67-74, 2014.

2. 2.  A semi-implicit augmented IIM for Navier-Stokes equations with open and traction boundary conditionsZ. Li, Q. Cai, H. Zhao, R. Luo, J. Comp. Phys. 297, 182-193, 2015.

3. 3.A cell based particle method for modeling dynamic interfaces, S. Y. Hon, S. Leung and H. Zhao, Journal of Computational Physics,  272, 279-306, 2014.

4. 4. Exploring Accurate Poisson-Boltzmann Methods for Biomolecular Simulations, C. Wang, J. Wang, Q. Cai, Z. Li, H. Zhao and R. Luo, Computational and Theoretical Chemistry, 1022:34–44, 2013.

5. 5. A weak formulation for solving elliptic interface problems without body fitted grid, S. Hou, P. Song, L. Wang and H. Zhao, J. Comp. Phys. Vol. 249, pp 80-95, 2013.

6. 6. Numerical Poisson--Boltzmann Model for Continuum Membrane Systems,  W. M. Botello-Smith, X. Liu, Q. Cai, Z. Li, H. Zhao, and Ray Luo, Chem. Phys. Lett., 555:274–281, 2013.

7. 7. Exploring a Charge-Central Strategy in the Solution of Poisson Equation for Biomolecular Applications, X. Liu, J. Wang, Z. Li, H. Zhao, and R. Luo, Phys. Chem. Chem. Phys.,15:129-141, 2013.

8. 8. A grid based particle method for high order geometrical motions and local inextensible flows, S. Leung, J. Lowengrub, and H. Zhao, J. Comp. Phys.  Vol. 230(7),  pp 2540-2561, 2011.

9. 9. Numerical study of surfactant-laden drop-drop interactionsJ. Xu, Z. Li, J. Lowengrub, and H. Zhao, Communications in Computational Physics, Vol. 10 (2), pp. 453-473, 2011.

10. 10. An augmented method for free boundary problems with moving contact lines, Z. Li, M.C. Lai, G. He, and H. Zhao, Computers & Fluids, 39, 1033-1040, 2010.

11. 11. Gaussian beam summation for diffraction in inhomogeneous media based on the grid based particle methodS. Leung and H. Zhao, Communication in Computational Physics. Vol. 8(4), pp 758-796, 2010.

12. 12. A Grid Based Particle Method for Evolution of Open Curves and Surfaces, S. Leung and H. Zhao, J. Comp. Phys. Vol., 228(20), pp 7706-7728, 2009.

13. 13. On removal of charge singularity in Poisson-Boltzmann equation, Q. Cai, J. Wang, H. Zhao, and R. Luo, Journal of Chemical Physics, 130:145101, 2009.

14. 14. Achieving energy conservation in Poisson-Boltzmann molecular dynamics: accuracy and precision with finite-difference algorithms, J. Wang, Q. Cai, Z.-L. Li, H. Zhao, and R. Luo, Chemical Physics Letters, 468:112-118, 2009.

15. 15. A novel grid based particle method for moving interface problem, S. Leung and H. Zhao, J. Comp. Phys., Vol. 228 (8), pp. 2993-3024, 2009.

16. 16. A Level set method for interfacial flows with surfactant, J. Xu, Z. Li, J. Lowengrub, H. Zhao, J. Comp. Phys. Vol. 212(2), pp. 590-616, 2006.

17. 17. Generalized Snell's law for weighted minimal surface in heterogeneous media, Z.  Li,  X.  Lin,  M. Torres, and H. Zhao, Methods and Applications of Analysis, Vol. 10 (2), 2003.

18. 18. An Eulerian formulation for solving partial differential equations along a moving interface, J. Xu and H. Zhao, Journal of Scientific Computing, Vol. 19, 2003, pp. 573-594.

19. 19. Reactive autophobic spreading of drops, J.K. Hunter, Z. Li, and H. Zhao,  J. Comp. Phys. Vol. 183, pp. 335-366, 2002.

20. 20. The geometry of Wulff crystal shapes and its relations with Riemann problems, D. Peng, S. Osher, B. Merriman, and H. Zhao, Contemporary Mathematics, Vol. 238, AMS, Providence, pp 251-303, eds G.-Q Chen and E. DiBenedetto, 1999.

21. 21. A PDE based fast local level set method, D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, J. Comp. Phys. Vol. 155, pp 410-438, 1999.

22. 22. A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid, Z. Li, H. Zhao, and H. Gao, J. Comp. Phys. Vol. 152, pp 281-304, 1999.

23. 23. Capturing the behavior of bubbles and drops using the variational level set approach, H. Zhao, B. Merriman, S. Osher, and, L. Wang, J. Comp. Phys. Vol. 143, pp 495-518, 1998.

24. 24. A hybrid method for moving interface problems with application to the Hele-Shaw flow, T.Y. Hou, Z.L. Li, S. Osher, and H. Zhao, J. Comp. Phys. Vol. 134, pp 236-252, 1997.

25. 25. A variational level set approach to multiphase motion, H. Zhao, T.F. Chan, B. Merriman, and S. Osher, J. Comp. Phys. Vol. 127, pp 179-195, 1996.

26. Inverse Problems and Imaging

27. 26. A hybrid adaptive phase space method for reflection traveltime tomography, H. Zhao and Y. Zhong, submitted, arXiv:1803.02501

28. 27. Two FFT subspace-based optimization methods for electrical impedance tomography, Z. Wei, R. Chen, H. Zhao and X. Chen, submitted.

29. 28. 5D respiratory motion model based image reconstruction algorithm for 4D cone-beam computed tomography, J. Liu, X. Zhang, X. Zhang, H. Zhao, Y. Gao, D. Thomas, D. Low, and H. Gao,Inverse Problems (Highlights of 2015), 31(11), 2015.

30. 29. Determining Scattering Support of Anisotropic Acoustic Mediums and Obstacles, H. Liu, H. Zhao and C. Zou, Communications in Mathematical Sciences,13, no. 4, 987--1000, 2015.

31. 30. Quantitative fluorescence photoacoustic tomography, K. Ren and H. Zhao, SIAM Journal on Imaging Sciences,6 (4), 2404-2429, 2013.

32. 31. A direct imaging method for inverse scattering using the generalized Foldy-Lax formulation, G. Bao, K. Huang, P. Li, and H. Zhao, Contemp. Math., AMS,615, 49-70, 2014.

33. 32. An efficient algorithm for the generalized Foldy-Lax formulation, K. Huang, P. Li, and H. Zhao, J. Comp. Phys. Vol. 234, pp 376-398, 2013.

34. 33.  A hybrid reconstruction method for quantitative photoacoustic tomography, K. Ren, H. Gao, and H. Zhao, SIAM Journal on Imaging Sciences, Vol. 6 (1), pp 32-55, 2013.

35. 34.Quantitative photoacoustic tomography, H. Gao, S. Osher, and H. Zhao, Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies, Lecture Notes in Mathematics: Mathematical Biosciences Subseries, Volume 2035, Springer-Verlag, Berlin, 2011.

36. 35. Adaptive phase space method with application to reflection traveltime tomography, E. Chung, J. Qian, G. Uhlmann, and H. Zhao, Inverse Problem, Inverse Problem, 27, 2011.

37. 36. An efficient Neumann-series based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, J. Qian, P. Stefanov, G. Uhlmann, and H. Zhao, SIAM Journal on Imaging Sciences, to appear.

38. 37. Bregman methods in quantitative photoacoustic tomography, H. Gao, S. Osher, and H. Zhao, Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies, Lecture Notes in Mathematics: Mathematical Biosciences Subseries, Volume 2035, Springer-Verlag, Berlin, 2011.

39. 38. Analysis of a fast forward solver for radiative transfer equation, H. Gao and H. Zhao, Mathematics of Computation, Vol. 82, pp 153-172, 2012..

40. 39. Bioluminescence tomography with Gaussian prior, H. Gao, H. Zhao, W. Cong, and G. Wang, Biomedical Optics Express, 1, pp 1259-1277, 2010.

41. 40. Fully linear reconstruction method for fluorescence yield and lifetime through inverse complex-source formulation, H. Gao, Y. Lin, G. Gulsen, and H. Zhao, Optics Letters. 35 1899-1901, 2010.

42. 41. Generalized Foldy-Lax formulation, K. Huang, K. Solna, and H. Zhao, J. Comp. Phys. Vol. 229(12), pp 4544-4553, 2010.

43. 42. Multilevel bioluminescence tomography based on radiative transfer equation Part 2: total variation and l1 data fidelity, H. Gao and H. Zhao, Optics Express. 18, 2894-2912, 2010.

44. 43. Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization, H. Gao and H. Zhao, Optics Express. 18, pp 1854-1871, 2010.

45. 44. A multilevel and multigrid optical tomography based on radiative transfer equation, H. Gao and H. Zhao, Proceedings of the SPIE, Vol. 7369, 73690E-73690E-10, 2009.

46. 45.  A fast forward solver of radiative transfer equation in optical imaging, H. Gao and H. Zhao, Transport Theory and Statistical Physics, Vol. 38(3)009 , pp 149-192, 2009.

47. 46. A phase and space coherent direct imaging algorithm, S. Hou, K. Huang, K. Solna, and H. Zhao, Journal of the Acoustical Society of America, Vol 125 (1), pp 227-238, 2009.

48. 47. Time reversal based direct imaging methods, H. Zhao, Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Y. Censor, M. Jiang and A.K. Louis (Editors), Edizioni della Normale, pp. 505-521, 2008.

49. 48. A phase space formulation for elastic-wave traveltime tomography, E. Chung, J. Qian, G. Uhlmann, and H. Zhao, Journal of Physics: Conference Series 124(2008): 012018.

50. 49. A Direct imaging method for far field data, S. Hou, K. Solna, and H. Zhao, Inverse Problem, Vol. 23, 1533-1546, 2007.

51. 50. A new phase space method for recovering index of refraction from traveltime, E. Chung, J. Qian, G. Uhlmann, and H. Zhao, Inverse Problems, Vol. 23, No. 1, pp. 309-329, 2007.

52. 51. A direct imaging algorithm for extended targets, S. Hou, K. Solna, H. Zhao, Inverse Problem, Vol. 22, 1151-1178, 2006.

53. 52. Imaging of location and geometry for extended targets using the response matrix, S. Hou, K. Solna, and H. Zhao, J. Comp. Phys. Vol. 199 (1), pp. 317-338, 2004.

54. 53. Analysis of the response matrix for an extended target, H. Zhao, SIAM Applied Mathematics, Vol. 64 No. 3, pp. 725-745. 2004.

55. 54. Super-resolution in time reversal acoustics, P. Blomgren, G. Papanicolaou, and H. Zhao, Journal of the Acoustical Society of America, Vol 111, pp. 230-248, 2002.

56. Image Processing and Computer Vision

1. 55. Neural Response Based Extreme Learning Machine for Image Classification, H. F. Li, H, Zhao and H. Li, IEEE Transactions on Neural Networks and Learning Systems, accepted..

2. 56. Modified virtual grid difference for discretizing Laplace-Beltrami operator on point clouds, M. Wang, S. Leung and H. Zhao, SIAM Journal on Scientific Computing, to appear.

3. 57. Real-time adaptive video compression, H. Schaeffer, Y. Yang, H. Zhao and S. Osher,SIAM Journal of Scientific Computing, 37(6), 980-1001, 2015.

4. 58. Multi-scale Non-Rigid Point Cloud Registration Using Robust Sliced-Wasserstein Distance via Laplace-Beltrami Eigenmap, R. Lai and H. Zhao, SIAM Journal on Imaging Sciences, 10(2), 449-483, 2017.

5. 59. Computation of Surface Uniformization Using Discrete Beltrami Flow, W. Wong and H. Zhao, SIAM Journal on Scientific Computing, 37(3), 1342-1364, 2015.

6. 60. Computation of quasiconformal surface maps using discrete Beltrami flow, T. W. Wong and H. Zhao, SIAM Journal on Imaging Sciences, 7(4), 2675–2699, 2014.

7. 61. Partially Blind Deblurring of Barcode from Out-of-Focus Blur, Y. Lou, E. Esser, H. Zhao and J. Xin, SIAM Journal on Imaging Sciences 7 (2), 740-760, 2014.

8. 62. Cine cone beam CT reconstruction using low-rank matrix factorization: algorithm and a proof-of-princple study, J.-F. Cai, X. Jia, H. Gao, S.B. Jiang, Z. Shen and H. Zhao, IEEE Transactions on Medical Imaging, 33(8):1581--1591, 2014.

9. 63. A local mesh method for solving PDEs on point clouds, R. Lai, J. Liang and H. Zhao, Inverse Problem and Imaging,7(3), 2013.

10. 64. Solving partial differential equations on point clouds, J. Liang and H. Zhao, SIAM Journal of Scientific Computing, 35(3), pp 1461-1486, 2013.

11. 65. Geometric Understanding of Point Clouds Using Laplace-Beltrami Operator, R. Lai, J. Liang, A. Wong and H. Zhao, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2012.

12. 66. Point cloud segmentation and denoising via constrained least squares normal estimates, E. Castillo, J. Liang, and H. Zhao, Book Chapter, Innovations for Shape Analysis: Models and Algorithms, Springer, 2012.

13. 67. Point cloud segmentation and denoising via constrained least squares normal estimates, E. Castillo, J. Liang and H. Zhao, Book Chapter, Innovations for Shape Analysis: Models and Algorithms, Springer, 2012.

14. 68. Robust principle component analysis based four-dimensional computed tomography, H. Gao, J. Cai, Z. Shen, and H. Zhao, Physics in Medicine and Biology, Vol. 56 (11), 3181. (featured article & Editor’s Choice)

15. 69. Euclidean skeletons using closest points, S. Luo, L. Guibas, and H. Zhao, Inverse Problems and Imaging, Vol. 30 (1), pp 95-113, 2011.

16. 70. A Nonparametric approach for noisy point data preprocessing, Y. Xi, Y. Duan, and H. Zhao, International Journel of CAD/CAM, Vol. 9, No. 1, pages. 31-36, 2009.

17. 71. Expectation-Maximization algorithm with local adaptivity for image analysis, S. Leung, G. Liang, K. Solna, and H. Zhao, SIAM Journal on Imaging Sciences, Vol. 2(3), pp 834-857, 2009.

18. 72. A surface reconstruction method for highly noisy point clouds, D. Lu, H. Zhao, M. Jiang, S. Zhou, and T. Zhou, N. Paragios et al. (Eds.): VLSM 2005. Lecture Notes in Computer Science, Springer, 3752, pp. 283-294, 2005.

19. 73. Swept volumes, M.  Peternell, H.  Pottmann, T. Steiner, and H. Zhao, Computer-Aided Design and Applications, Vol. 2, No. 5, 2005.

20. 74. Visualization, analysis and shape reconstruction of unorganized data sets, H. Zhao and S. Osher, book chapter in Geometric Level Set Methods in Imaging, Vision and Graphics, S. Osher and N. Paragios Editors,Springer, 2003.

21. 75. Fast surface reconstruction and deformation using the level set method, H. Zhao, S. Osher, and R. Fedkiw, Proceedings of IEEE Workshop on Variational and Level Set Methods in Computer Vision (VLSM  2001), Vancouver, 2001.

22. 76. Implicit and non-parametric shape reconstruction from unorganized points using variational level set method, H. Zhao, S. Osher, B. Merriman, and M. Kang, Computer Vision and Image Understanding. Vol. 80, pp 295-319, 2000.

23. Hamilton-Jacobi Equation and the Fast Sweeping Method (FSM)

24. 77. The fast sweeping method for stationary Hamilton-Jacobi equations, Handbook of Numerical Methods for Hyperbolic Problems, Basics and Fundamental Issues, Vol. 17, Ed. R. Abgrall and C.W. Shu, Elsevier, 2016.

25. 78.Convergence analysis of the fast sweeping method for static convex Hamilton-Jacobi equation, S. Luo and H. Zhao, Research in Mathematical Sciences, 3:35, 2016.

26. 79. A static PDE approach for multi-dimensional extrapolation using fast sweeping methods, T. Aslam, S. Luo and H. Zhao,  SIAM Journal on Scientific Computing, 36(6), 2014.

27. 80. Higher-order schemes for 3-D traveltimes and amplitudes, S. Luo, J. Qian, and H. Zhao, Geophysics, Vol 77 (2), pp 47-56, 2012.

28. 81. Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations, Y.-T. Zhang, S. Chen, F. Li, H. Zhao, and C.-W. Shu, SIAM Journal of Scientific Computing, 33(4), pp 1873-1896, 2011.

29. 82. A new approximation for effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations, S. Luo, Y. Yu, and H. Zhao, SIAM Journal on Multiscale Modeling and Simulation, 9(2), pp 711-734, 2011.

30. 83. A compact upwind second order scheme for the Eikonal equation, J. D. Benamou, S. Luo and H. Zhao, Journal of Computational Mathematics, 28, pp 489-516, 2010.

31. 84. Fast sweeping method for the factored eikonal equation, S. Fomel, S. Luo and H. Zhao, J. Comp. Phys., Vol. 228(17), pp 6440-6455, 2009.

32. 85.  Second order discontinuous Galerkin fast sweeping method for Eikonal equations,  F. Li, C.-W. Shu, Y.-T. Zhang, and  H. Zhao, J. Comp. Phys., Vol. 227(17), pp. 8191-8208, 2008.

33. 86. A fast sweeping method for static convex Hamilton-Jacobi equations, J. Qian, Y. Zhang, and H. Zhao, Journal of Scientific Computing, Vol. 31, No. 1/2, pp. 237-271, 2007.

34. 87. Parallel implementation of fast sweeping method, H. Zhao, Journal of Computational Mathematics, Vol. 25, No. 4, pp. 421-429, 2007.

35. 88. Fast sweeping methods for Eikonal equations on triangulated meshes, J. Qian, Y. Zhang, and H. Zhao,  SIAM Journal on Numerical Analysis, Vol. 45, pp. 83-107, 2007.

36. 89. Fixed-point iterative sweeping methods for steady-states of Hamilton-Jacobi equations, Y. Zhang, H. Zhao, and S. Chen, Methods and Applications of Analysis Vol. 13, pp. 299-320, 2006.

37. 90. High order fast sweeping methods for static Hamilton-Jacobi equations, Y. Zhang,  H. Zhao, and J. Qian, Journal of Scientific Computing, Vol. 29, pp. 25-56, 2006.

38. 91. Fast sweeping method for eikonal equations, H. Zhao, Mathematics of Computation, Vol. 74, 603-627, 2005.

39. 92. Fast sweeping algorithms for a class of Hamilton-Jacobi equations, Y.R. Tsai,  L.T. Cheng,  S. Osher, and H. Zhao, SIAM  Journal on Numerical Analysis, Vol 41, No 2, pp. 673-694, 2003.

Others

1. 93. Intrinsic Complexity And Scaling Laws: From Random Fields to Random Vectors, J. Bryson, H. Zhao and Y. Zhong, submitted. arXiv:1805.00194

2. 94. An efficient hybrid method for high frequency Helmholtz equation with point source, . Fang, J. Qian, L. Zepeda-Nunez and H. Zhao, J. Comp. Phys., accepted.

3. 95. Approximate Separability of Green's Function for High Frequency Helmholtz Equations, B. Engquist and H. Zhao, Communications on Pure and Applied Mathematics, accepted.

4. 96. Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations, J. Fang, J. Qian, L. Zepeda-Nunez and H. Zhao, Research in Mathematical Sciences, 4(12), 2017.

5. 97. Fast alternating bi-directional preconditioner for the 2D high frequency Lippmann-Schwinger equation, L. Zepeda-Nunez and H. Zhao, SIAM Journal on Scientific Computing, 38(5), 866-888, 2017..

6. 98. Variational Hamiltonian Monte Carlo via score matching, C. Zhang, B. Shahbaba and H. Zhao, Bayesian Analysis, to appear.

7. 99. Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases, C. Zhang, B. Shahbaba and H. Zhao, Statistics and Computing, to appear.

8. 100. Precomputing strategy for Hamiltonian Monte Carlo method based on regularity in parameter space, C. Zhang, B. Shahbaba and H. Zhao, Computational Statistics, 32(1), 253-279, 2017.

9. 101. Efficient numerical simulation for long range wave propagation, K. Huang, G. Papanicolaou, K. Solna, C. Tsogka, and H. Zhao,  J. Comp. Phys. Vol. 215(2), pp. 448-464, 2006.

10. 102. Coupled parabolic equations for wave propagation, K. Huang, K. Solna, and H. Zhao,  Methods and Applications of Analysis, Vol. 11 (3), pp 399-412, 2004.

11. 103.Overlapping Schwarz Waveform Relaxation for the Heat Equation in n-Dimensions, M.J. Gander and H. Zhao, BIT, Vol. 42, No. 4, pp. 779-795, 2002.

12. 104. Absorbing boundary conditions for domain decomposition, B. Engquist and H. Zhao, Applied Numerical Mathematics, Vol. 27, pp341-365, 1998.